Compute the long run probability distribution over states if an equilibrium is played for many periods.

Adds a column state.prob to the computed equilibrium data frame, which you can retrieve by calling get_eq.

rel_state_probs(
  g,
  x0 = c("equal", "first", "first.group")[1],
  start.prob = NULL,
  n.burn.in = 100,
  n.averaging = 100,
  tol = 1e-13,
  eq.field = "eq"
)

Arguments

g	the game object for which an equilibrium has been solved
x0	the initial state, by default the first state. If initial is not specified we asssume the game starts with probability 1 in the initial state. We have 3 reserved keywords: x0="equal" means all states are equally likely, x0="first" means the game starts in the first state, x0="first.group" means all states of the first xgroup are equally likely.
start.prob	an optional vector of probabilities that specifies for each state the probability that the game starts in that state. Overwrites "x0" unless kept NULL.
n.burn.in	Number of rounds before probabilities are averaged.
n.averaging	Number of rounds for which probabilities are averaged.
tol	Tolerance such that computation stops already in burn-in phase if transition probabilities change not by more than tol.

Details

If the equilibrium strategy induces a unique stationary distribution over the states, this distribution should typically be found (or at least approximated). Otherwise the result can depend on the parameters.

The initial distribution of states is determined by the parameters x0 or start.prob. We then multiply the current probabilities susequently n.burn.in times with the transitition matrix on the equilibrium path. This yields the probability distribution over states assuming the game is played for n.burn.in periods.

We then continue the process for n.averaging rounds, and return the mean of the state probability vectors over these number of rounds.

If between two rounds in the burn-in phase the transitition probabilities of no state pair change by more than the parameter tol, we immediately stop and use the resulting probabilit vector.

Note that for T-RNE or capped RNE we always take the transition probabilities of the first period, i.e. we don't increase the t in the actual state definition.